Base loses crispness (or turns soggy) while resting after baking

Cosmetic

Almost always trapped steam from resting on a sealed or solid surface, not a fault in the bake itself.

Ask yourself

#1
Steam trapped underneath the pizza by a sealed box, flat plate, or other non-vented surface
Strongly supported
A freshly baked crust keeps releasing steam for several minutes; trapped against a solid surface, that steam condenses back into the base and softens it.
#2
Residual moisture from sauce or toppings continuing to migrate into the crust after baking stops
Generally accepted
Moisture doesn't stop moving once the pizza leaves the oven; a heavily sauced or topped pizza keeps losing crispness for longer than a lighter one.

The same symptom has different answers on different equipment. Each branch below states how to tell whether it is yours.

Pizza in a domestic ovenGenerally accepted
How you would know: The base was crisp out of the oven and soft within a minute or two of sitting on a board or plate.
Steam trapped underneath. A hot pizza continues to release water vapour, and on a flat surface that vapour condenses against the base and re-hydrates the crust you just dried.
Do now: Move it onto a wire rack immediately, even mid-slice.
Change next time: Rest every pizza on a rack rather than a board, and cut it there. This undoes a genuinely good bake more often than people realise, and costs nothing to fix.

A brief return to a very hot oven, or a few minutes directly on a dry, hot pan or skillet, re-crisps a softened base

Rest the pizza on a wire rack or other vented surface rather than a sealed box or flat plate, and slice and serve reasonably promptly

Guides that treat this as an outcome to improve, an environment to work with, or a flavour to understand, rather than only a problem to fix.